### The Mizar article:

### Subspaces and Cosets of Subspaces in Real Linear Space

**by****Wojciech A. Trybulec**

- Received July 24, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: RLSUB_1
- [ MML identifier index ]

environ vocabulary RLVECT_1, BOOLE, ARYTM_1, RELAT_1, FUNCT_1, BINOP_1, RLSUB_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, REAL_1, MCART_1, FUNCT_1, RELSET_1, FUNCT_2, DOMAIN_1, BINOP_1, STRUCT_0, RLVECT_1; constructors REAL_1, DOMAIN_1, RLVECT_1, PARTFUN1, MEMBERED, XBOOLE_0; clusters FUNCT_1, RLVECT_1, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, BOOLE, SUBSET, ARITHM; definitions RLVECT_1, TARSKI, XBOOLE_0; theorems FUNCT_1, FUNCT_2, RLVECT_1, TARSKI, ZFMISC_1, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1; schemes XBOOLE_0; begin reserve V,X,Y for RealLinearSpace; reserve u,u1,u2,v,v1,v2 for VECTOR of V; reserve a,b for Real; reserve V1,V2,V3 for Subset of V; reserve x for set; :: :: Introduction of predicate lineary closed subsets of the carrier. :: definition let V; let V1; attr V1 is lineary-closed means :Def1: (for v,u st v in V1 & u in V1 holds v + u in V1) & (for a,v st v in V1 holds a * v in V1); end; canceled 3; theorem Th4: V1 <> {} & V1 is lineary-closed implies 0.V in V1 proof assume that A1: V1 <> {} and A2: V1 is lineary-closed; consider x being Element of V1; reconsider x as Element of V by A1,TARSKI:def 3; 0 * x in V1 by A1,A2,Def1; hence thesis by RLVECT_1:23; end; theorem Th5: V1 is lineary-closed implies (for v st v in V1 holds - v in V1) proof assume A1: V1 is lineary-closed; let v; assume v in V1; then (- 1) * v in V1 by A1,Def1; hence thesis by RLVECT_1:29; end; theorem V1 is lineary-closed implies (for v,u st v in V1 & u in V1 holds v - u in V1) proof assume A1: V1 is lineary-closed; let v,u; assume that A2: v in V1 and A3: u in V1; v - u = v + (- u) & - u in V1 by A1,A3,Th5,RLVECT_1:def 11; hence thesis by A1,A2,Def1; end; theorem Th7: {0.V} is lineary-closed proof thus for v,u st v in {0.V} & u in {0.V} holds v + u in {0.V} proof let v,u; assume v in {0.V} & u in {0.V}; then v = 0.V & u = 0.V by TARSKI:def 1; then v + u = 0.V & 0.V in {0.V} by RLVECT_1:10,TARSKI:def 1; hence thesis; end; let a,v; assume A1: v in {0.V}; then v = 0.V by TARSKI:def 1; hence thesis by A1,RLVECT_1:23; end; theorem the carrier of V = V1 implies V1 is lineary-closed proof assume A1: the carrier of V = V1; hence for v,u st v in V1 & u in V1 holds v + u in V1; let a,v; assume v in V1; thus a * v in V1 by A1; end; theorem V1 is lineary-closed & V2 is lineary-closed & V3 = {v + u : v in V1 & u in V2} implies V3 is lineary-closed proof assume that A1: V1 is lineary-closed & V2 is lineary-closed and A2: V3 = {v + u : v in V1 & u in V2}; thus for v,u st v in V3 & u in V3 holds v + u in V3 proof let v,u; assume that A3: v in V3 and A4: u in V3; consider v1,v2 such that A5: v = v1 + v2 and A6: v1 in V1 & v2 in V2 by A2,A3; consider u1,u2 such that A7: u = u1 + u2 and A8: u1 in V1 & u2 in V2 by A2,A4; A9: v1 + u1 in V1 & v2 + u2 in V2 by A1,A6,A8,Def1; v + u = ((v1 + v2) + u1) + u2 by A5,A7,RLVECT_1:def 6 .= ((v1 + u1) + v2) + u2 by RLVECT_1:def 6 .= (v1 + u1) + (v2 + u2) by RLVECT_1:def 6; hence thesis by A2,A9; end; let a,v; assume v in V3; then consider v1,v2 such that A10: v = v1 + v2 and A11: v1 in V1 & v2 in V2 by A2; A12: a * v1 in V1 & a * v2 in V2 by A1,A11,Def1; a * v = a * v1 + a * v2 by A10,RLVECT_1:def 9; hence a * v in V3 by A2,A12; end; theorem V1 is lineary-closed & V2 is lineary-closed implies V1 /\ V2 is lineary-closed proof assume A1: V1 is lineary-closed & V2 is lineary-closed; thus for v,u st v in V1 /\ V2 & u in V1 /\ V2 holds v + u in V1 /\ V2 proof let v,u; assume v in V1 /\ V2 & u in V1 /\ V2; then v in V1 & v in V2 & u in V1 & u in V2 by XBOOLE_0:def 3; then v + u in V1 & v + u in V2 by A1,Def1; hence thesis by XBOOLE_0:def 3; end; let a,v; assume v in V1 /\ V2; then v in V1 & v in V2 by XBOOLE_0:def 3; then a * v in V1 & a * v in V2 by A1,Def1; hence thesis by XBOOLE_0:def 3; end; definition let V; mode Subspace of V -> RealLinearSpace means :Def2: the carrier of it c= the carrier of V & the Zero of it = the Zero of V & the add of it = (the add of V) | [:the carrier of it,the carrier of it:] & the Mult of it = (the Mult of V) | [:REAL, the carrier of it:]; existence proof the carrier of V c= the carrier of V & the Zero of V = the Zero of V & the add of V = (the add of V) | [:the carrier of V,the carrier of V:] & the Mult of V = (the Mult of V) | [:REAL, the carrier of V:] by FUNCT_2:40 ; hence thesis; end; end; reserve W,W1,W2 for Subspace of V; reserve w,w1,w2 for VECTOR of W; :: :: Axioms of the subspaces of real linear spaces. :: canceled 5; theorem x in W1 & W1 is Subspace of W2 implies x in W2 proof assume x in W1 & W1 is Subspace of W2; then x in the carrier of W1 & the carrier of W1 c= the carrier of W2 by Def2,RLVECT_1:def 1 ; hence thesis by RLVECT_1:def 1; end; theorem Th17: x in W implies x in V proof assume x in W; then x in the carrier of W & the carrier of W c= the carrier of V by Def2,RLVECT_1:def 1 ; hence thesis by RLVECT_1:def 1; end; theorem Th18: w is VECTOR of V proof w in W by RLVECT_1:3; then w in V by Th17; hence thesis by RLVECT_1:def 1; end; theorem Th19: 0.W = 0.V proof thus 0.W = the Zero of W by RLVECT_1:def 2 .= the Zero of V by Def2 .= 0.V by RLVECT_1:def 2; end; theorem 0.W1 = 0.W2 proof thus 0.W1 = 0.V by Th19 .= 0.W2 by Th19; end; theorem Th21: w1 = v & w2 = u implies w1 + w2 = v + u proof assume A1: v = w1 & u = w2; reconsider ww1 = w1, ww2 = w2 as VECTOR of V by Th18; A2: v + u = (the add of V).[ww1,ww2] by A1,RLVECT_1:def 3; w1 + w2 = (the add of W).[w1,w2] by RLVECT_1:def 3 .= ((the add of V) | [:the carrier of W, the carrier of W:] ).[w1,w2] by Def2; hence thesis by A2,FUNCT_1:72; end; theorem Th22: w = v implies a * w = a * v proof assume A1: w = v; reconsider ww1 = w as VECTOR of V by Th18; A2: a * v = (the Mult of V).[a,ww1] by A1,RLVECT_1:def 4; a * w = (the Mult of W).[a,w] by RLVECT_1:def 4 .= ((the Mult of V) | [:REAL, the carrier of W:]).[a,w] by Def2; hence thesis by A2,FUNCT_1:72; end; theorem Th23: w = v implies - v = - w proof assume A1: w = v; - v = (- 1) * v & - w = (- 1) * w by RLVECT_1:29; hence thesis by A1,Th22; end; theorem Th24: w1 = v & w2 = u implies w1 - w2 = v - u proof assume that A1: w1 = v and A2: w2 = u; A3: - w2 = - u by A2,Th23; w1 - w2 = w1 + (- w2) & v - u = v + (- u) by RLVECT_1:def 11; hence thesis by A1,A3,Th21; end; Lm1: the carrier of W = V1 implies V1 is lineary-closed proof assume A1: the carrier of W = V1; set VW = the carrier of W; reconsider WW = W as RealLinearSpace; thus for v,u st v in V1 & u in V1 holds v + u in V1 proof let v,u; assume v in V1 & u in V1; then reconsider vv = v, uu = u as VECTOR of WW by A1; reconsider vw = vv + uu as Element of VW; vw in V1 by A1; hence v + u in V1 by Th21; end; let a,v; assume v in V1; then reconsider vv = v as VECTOR of WW by A1; reconsider vw = a * vv as Element of VW; vw in V1 by A1; hence a * v in V1 by Th22; end; theorem Th25: 0.V in W proof 0.W in W & 0.V = 0.W by Th19,RLVECT_1:3; hence thesis; end; theorem 0.W1 in W2 proof 0.W1 = 0.V by Th19; hence thesis by Th25; end; theorem 0.W in V proof 0.W in W by RLVECT_1:3; hence thesis by Th17; end; theorem Th28: u in W & v in W implies u + v in W proof assume u in W & v in W; then A1: u in the carrier of W & v in the carrier of W by RLVECT_1:def 1; reconsider VW = the carrier of W as Subset of V by Def2; VW is lineary-closed by Lm1; then u + v in the carrier of W by A1,Def1; hence thesis by RLVECT_1:def 1; end; theorem Th29: v in W implies a * v in W proof assume v in W; then A1: v in the carrier of W by RLVECT_1:def 1; reconsider VW = the carrier of W as Subset of V by Def2; VW is lineary-closed by Lm1; then a * v in the carrier of W by A1,Def1; hence thesis by RLVECT_1:def 1; end; theorem Th30: v in W implies - v in W proof assume v in W; then (- 1) * v in W by Th29; hence thesis by RLVECT_1:29; end; theorem Th31: u in W & v in W implies u - v in W proof assume that A1: u in W and A2: v in W; - v in W by A2,Th30; then u + (- v) in W by A1,Th28; hence thesis by RLVECT_1:def 11; end; reserve D for non empty set; reserve d1 for Element of D; reserve A for BinOp of D; reserve M for Function of [:REAL,D:],D; theorem Th32: V1 = D & d1 = 0.V & A = (the add of V) | [:V1,V1:] & M = (the Mult of V) | [:REAL,V1:] implies RLSStruct (# D,d1,A,M #) is Subspace of V proof assume that A1: V1 = D and A2: d1 = 0.V and A3: A = (the add of V) | [:V1,V1:] and A4: M = (the Mult of V) | [:REAL,V1:]; set W = RLSStruct (# D,d1,A,M #); A5: the Zero of W = the Zero of V by A2,RLVECT_1:def 2; A6: for x,y being VECTOR of W holds x + y = (the add of V).[x,y] proof let x,y be VECTOR of W; x + y = ((the add of V) | [:the carrier of W, the carrier of W:] ).[x,y] by A1,A3,RLVECT_1:def 3; hence thesis by FUNCT_1:72; end; A7: for a for x being VECTOR of W holds a * x = (the Mult of V).[a,x] proof let a; let x be VECTOR of W; a * x = ((the Mult of V) | [:REAL, the carrier of W:] ).[a,x] by A1,A4,RLVECT_1:def 4; hence thesis by FUNCT_1:72; end; A8: d1 = 0.W by RLVECT_1:def 2; W is Abelian add-associative right_zeroed right_complementable RealLinearSpace-like proof set AV = the add of V; set MV = the Mult of V; thus for x,y being VECTOR of W holds x + y = y + x proof let x,y be VECTOR of W; reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3; thus x + y = AV.[x1,y1] by A6 .= y1 + x1 by RLVECT_1:def 3 .= AV.[y1,x1] by RLVECT_1:def 3 .= y + x by A6; end; thus for x,y,z being VECTOR of W holds (x + y) + z = x + (y + z) proof let x,y,z be VECTOR of W; reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1,TARSKI:def 3; thus (x + y) + z = AV.[x + y,z1] by A6 .= AV.[AV.[x1,y1],z1] by A6 .= AV.[x1 + y1,z1] by RLVECT_1:def 3 .= (x1 + y1) + z1 by RLVECT_1:def 3 .= x1 + (y1 + z1) by RLVECT_1:def 6 .= AV.[x1,y1 + z1] by RLVECT_1:def 3 .= AV.[x1,AV.[y1,z1]] by RLVECT_1:def 3 .= AV.[x1,y + z] by A6 .= x + (y + z) by A6; end; thus for x being VECTOR of W holds x + 0.W = x proof let x be VECTOR of W; reconsider y = x, z = 0.W as VECTOR of V by A1,TARSKI:def 3; thus x + 0.W = AV.[y,z] by A6 .= y + 0.V by A2,A8,RLVECT_1:def 3 .= x by RLVECT_1:10; end; thus for x being VECTOR of W ex y being VECTOR of W st x + y = 0.W proof let x be VECTOR of W; reconsider x1 = x as VECTOR of V by A1,TARSKI:def 3; consider v such that A9: x1 + v = 0.V by RLVECT_1:def 8; v = - x1 by A9,RLVECT_1:def 10 .= (- 1) * x1 by RLVECT_1:29 .= MV.[- 1,x1] by RLVECT_1:def 4 .= (- 1) * x by A7; then reconsider y = v as VECTOR of W; take y; thus x + y = AV.[x1,v] by A6 .= 0.W by A2,A8,A9,RLVECT_1:def 3; end; thus for a for x,y being VECTOR of W holds a * (x + y) = a * x + a * y proof let a; let x,y be VECTOR of W; reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3; thus a * (x + y) = MV.[a,x + y] by A7 .= MV.[a,AV.[x1,y1]] by A6 .= MV.[a,x1 + y1] by RLVECT_1:def 3 .= a * (x1 + y1) by RLVECT_1:def 4 .= a * x1 + a * y1 by RLVECT_1:def 9 .= AV.[a * x1,a * y1] by RLVECT_1:def 3 .= AV.[MV.[a,x1],a * y1] by RLVECT_1:def 4 .= AV.[MV.[a,x1],MV.[a,y1]] by RLVECT_1:def 4 .= AV.[MV.[a,x1],a * y] by A7 .= AV.[a * x, a * y] by A7 .= a * x + a * y by A6; end; thus for a,b for x being VECTOR of W holds (a + b) * x = a * x + b * x proof let a,b; let x be VECTOR of W; reconsider y = x as VECTOR of V by A1,TARSKI:def 3; thus (a + b) * x = MV.[a + b,y] by A7 .= (a + b) * y by RLVECT_1:def 4 .= a * y + b * y by RLVECT_1:def 9 .= AV.[a * y,b * y] by RLVECT_1:def 3 .= AV.[MV.[a,y],b * y] by RLVECT_1:def 4 .= AV.[MV.[a,y],MV.[b,y]] by RLVECT_1:def 4 .= AV.[MV.[a,y],b * x] by A7 .= AV.[a * x,b * x] by A7 .= a * x + b * x by A6; end; thus for a,b for x being VECTOR of W holds (a * b) * x = a * (b * x) proof let a,b; let x be VECTOR of W; reconsider y = x as VECTOR of V by A1,TARSKI:def 3; thus (a * b) * x = MV.[(a * b),y] by A7 .= (a * b) * y by RLVECT_1:def 4 .= a * (b * y) by RLVECT_1:def 9 .= MV.[a,b * y] by RLVECT_1:def 4 .= MV.[a,MV.[b,y]] by RLVECT_1:def 4 .= MV.[a,b * x] by A7 .= a * (b * x) by A7; end; let x be VECTOR of W; reconsider y = x as VECTOR of V by A1,TARSKI:def 3; thus 1 * x = MV.[1,y] by A7 .= 1 * y by RLVECT_1:def 4 .= x by RLVECT_1:def 9; end; hence thesis by A1,A3,A4,A5,Def2; end; theorem Th33: V is Subspace of V proof thus the carrier of V c= the carrier of V & the Zero of V = the Zero of V; thus thesis by FUNCT_2:40; end; theorem Th34: for V,X being strict RealLinearSpace holds V is Subspace of X & X is Subspace of V implies V = X proof let V,X be strict RealLinearSpace; assume A1: V is Subspace of X & X is Subspace of V; set VV = the carrier of V; set VX = the carrier of X; set AV = the add of V; set AX = the add of X; set MV = the Mult of V; set MX = the Mult of X; VV c= VX & VX c= VV by A1,Def2; then A2: VV = VX by XBOOLE_0:def 10; A3: the Zero of V = the Zero of X by A1,Def2; AV = AX | [:VV,VV:] & AX = AV | [:VX,VX:] by A1,Def2; then A4: AV = AX by A2,RELAT_1:101; MV = MX | [:REAL,VV:] & MX = MV | [:REAL,VX:] by A1,Def2; hence thesis by A2,A3,A4,RELAT_1:101; end; theorem Th35: V is Subspace of X & X is Subspace of Y implies V is Subspace of Y proof assume A1: V is Subspace of X & X is Subspace of Y; thus the carrier of V c= the carrier of Y proof the carrier of V c= the carrier of X & the carrier of X c= the carrier of Y by A1,Def2; hence thesis by XBOOLE_1:1; end; thus the Zero of V = the Zero of Y proof the Zero of V = the Zero of X & the Zero of X = the Zero of Y by A1,Def2; hence thesis; end; thus the add of V = (the add of Y) | [:the carrier of V, the carrier of V:] proof set AV = the add of V; set VV = the carrier of V; set AX = the add of X; set VX = the carrier of X; set AY = the add of Y; AV = AX | [:VV,VV:] & AX = AY | [:VX,VX:] & VV c= VX by A1,Def2; then AV = (AY | [:VX,VX:]) | [:VV,VV:] & [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:119; hence thesis by FUNCT_1:82; end; set MV = the Mult of V; set VV = the carrier of V; set MX = the Mult of X; set VX = the carrier of X; set MY = the Mult of Y; MV = MX | [:REAL,VV:] & MX = MY | [:REAL,VX:] & VV c= VX by A1,Def2; then MV = (MY | [:REAL,VX:]) | [:REAL,VV:] & [:REAL,VV:] c= [:REAL,VX:] by ZFMISC_1:118; hence thesis by FUNCT_1:82; end; theorem Th36: the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2 proof assume A1: the carrier of W1 c= the carrier of W2; set VW1 = the carrier of W1; set VW2 = the carrier of W2; set AV = the add of V; set MV = the Mult of V; the Zero of W1 = the Zero of V & the Zero of W2 = the Zero of V by Def2; hence the carrier of W1 c= the carrier of W2 & the Zero of W1 = the Zero of W2 by A1; thus the add of W1 = (the add of W2) | [:the carrier of W1,the carrier of W1:] proof the add of W1 = AV | [:VW1,VW1:] & the add of W2 = AV | [:VW2,VW2:] & [:VW1,VW1:] c= [:VW2,VW2:] by A1,Def2,ZFMISC_1:119; hence thesis by FUNCT_1:82; end; the Mult of W1 = MV | [:REAL,VW1:] & the Mult of W2 = MV | [:REAL,VW2:] & [:REAL,VW1:] c= [:REAL,VW2:] by A1,Def2,ZFMISC_1:118; hence thesis by FUNCT_1:82; end; theorem (for v st v in W1 holds v in W2) implies W1 is Subspace of W2 proof assume A1: for v st v in W1 holds v in W2; the carrier of W1 c= the carrier of W2 proof let x be set; assume A2: x in the carrier of W1; the carrier of W1 c= the carrier of V by Def2; then reconsider v = x as VECTOR of V by A2; v in W1 by A2,RLVECT_1:def 1; then v in W2 by A1; hence thesis by RLVECT_1:def 1; end; hence thesis by Th36; end; definition let V; cluster strict Subspace of V; existence proof the carrier of V is Subset of V iff the carrier of V c= the carrier of V; then reconsider V1 = the carrier of V as Subset of V; the Zero of V = 0.V & the add of V = (the add of V) | [:V1,V1:] & the Mult of V = (the Mult of V) | [:REAL,V1:] by FUNCT_2:40,RLVECT_1:def 2 ; then RLSStruct(#the carrier of V,the Zero of V,the add of V,the Mult of V #) is Subspace of V by Th32; hence thesis; end; end; theorem Th38: for W1,W2 being strict Subspace of V holds the carrier of W1 = the carrier of W2 implies W1 = W2 proof let W1,W2 be strict Subspace of V; assume the carrier of W1 = the carrier of W2; then W1 is Subspace of W2 & W2 is Subspace of W1 by Th36; hence thesis by Th34; end; theorem Th39: for W1,W2 being strict Subspace of V holds (for v holds v in W1 iff v in W2) implies W1 = W2 proof let W1,W2 be strict Subspace of V; assume A1: for v holds v in W1 iff v in W2; x in the carrier of W1 iff x in the carrier of W2 proof thus x in the carrier of W1 implies x in the carrier of W2 proof assume A2: x in the carrier of W1; the carrier of W1 c= the carrier of V by Def2; then reconsider v = x as VECTOR of V by A2; v in W1 by A2,RLVECT_1:def 1; then v in W2 by A1; hence thesis by RLVECT_1:def 1; end; assume A3: x in the carrier of W2; the carrier of W2 c= the carrier of V by Def2; then reconsider v = x as VECTOR of V by A3; v in W2 by A3,RLVECT_1:def 1; then v in W1 by A1; hence thesis by RLVECT_1:def 1; end; then the carrier of W1 = the carrier of W2 by TARSKI:2; hence thesis by Th38; end; theorem for V being strict RealLinearSpace, W being strict Subspace of V holds the carrier of W = the carrier of V implies W = V proof let V be strict RealLinearSpace, W be strict Subspace of V; assume A1: the carrier of W = the carrier of V; V is Subspace of V by Th33; hence thesis by A1,Th38; end; theorem for V being strict RealLinearSpace, W being strict Subspace of V holds (for v being VECTOR of V holds v in W iff v in V) implies W = V proof let V be strict RealLinearSpace, W be strict Subspace of V; assume A1: for v being VECTOR of V holds v in W iff v in V; V is Subspace of V by Th33; hence thesis by A1,Th39; end; theorem the carrier of W = V1 implies V1 is lineary-closed by Lm1; theorem Th43: V1 <> {} & V1 is lineary-closed implies (ex W being strict Subspace of V st V1 = the carrier of W) proof assume that A1: V1 <> {} and A2: V1 is lineary-closed; reconsider D = V1 as non empty set by A1; reconsider d1 = 0.V as Element of D by A2,Th4; set A = (the add of V) | [:V1,V1:]; set M = (the Mult of V) | [:REAL,V1:]; set VV = the carrier of V; dom(the add of V) = [:VV,VV:] by FUNCT_2:def 1; then dom A = [:VV,VV:] /\ [:V1,V1:] & [:V1,V1:] c= [:VV,VV:] by RELAT_1:90; then A3: dom A = [:D,D:] by XBOOLE_1:28; dom(the Mult of V) = [:REAL,VV:] by FUNCT_2:def 1; then dom M = [:REAL,VV:] /\ [:REAL,V1:] & [:REAL,V1:] c= [:REAL,VV:] by RELAT_1:90,ZFMISC_1:118; then A4: dom M = [:REAL,D:] by XBOOLE_1:28; A5: D = rng A proof now let y be set; thus y in D implies ex x being set st x in dom A & y = A.x proof assume A6: y in D; then reconsider v1 = y, v0 = d1 as Element of VV; A7: [d1,y] in [:D,D:] & [d1,y] in [:VV,VV:] by A6,ZFMISC_1:106; then A.[d1,y] = (the add of V).[d1,y] by FUNCT_1:72 .= v0 + v1 by RLVECT_1:def 3 .= y by RLVECT_1:10; hence thesis by A3,A7; end; given x being set such that A8: x in dom A and A9: y = A.x; consider x1,x2 being set such that A10: x1 in D & x2 in D and A11: x = [x1,x2] by A3,A8,ZFMISC_1:def 2; A12: [x1,x2] in [:VV,VV:] & [x1,x2] in [:V1,V1:] by A10,ZFMISC_1:106; reconsider v1 = x1, v2 = x2 as Element of VV by A10; y = (the add of V).[x1,x2] by A9,A11,A12,FUNCT_1:72 .= v1 + v2 by RLVECT_1:def 3; hence y in D by A2,A10,Def1; end; hence thesis by FUNCT_1:def 5; end; A13: D = rng M proof now let y be set; thus y in D implies ex x being set st x in dom M & y = M.x proof assume A14: y in D; then reconsider v1 = y as Element of VV; A15: [1,y] in [:REAL,D:] & [1,y] in [:REAL,VV:] by A14,ZFMISC_1:106; then M.[1,y] = (the Mult of V).[1,y] by FUNCT_1:72 .= 1 * v1 by RLVECT_1:def 4 .= y by RLVECT_1:def 9; hence thesis by A4,A15; end; given x being set such that A16: x in dom M and A17: y = M.x; consider x1,x2 being set such that A18: x1 in REAL and A19: x2 in D and A20: x = [x1,x2] by A4,A16,ZFMISC_1:def 2; A21: [x1,x2] in [:REAL,VV:] & [x1,x2] in [:REAL,V1:] by A18,A19,ZFMISC_1:106; reconsider v2 = x2 as Element of VV by A19; reconsider xx1 = x1 as Real by A18; y = (the Mult of V).[x1,x2] by A17,A20,A21,FUNCT_1:72 .= xx1 * v2 by RLVECT_1:def 4; hence y in D by A2,A19,Def1; end; hence thesis by FUNCT_1:def 5; end; reconsider A as Function of [:D,D:],D by A3,A5,FUNCT_2:def 1,RELSET_1:11; reconsider M as Function of [:REAL,D:],D by A4,A13,FUNCT_2:def 1,RELSET_1:11; set W = RLSStruct (# D,d1,A,M #); W is Subspace of V & the carrier of W = D by Th32; hence thesis; end; :: :: Definition of zero subspace and improper subspace of real linear space. :: definition let V; func (0).V -> strict Subspace of V means :Def3: the carrier of it = {0.V}; correctness proof {0.V} is lineary-closed & {0.V} <> {} by Th7; hence thesis by Th38,Th43; end; end; definition let V; func (Omega).V -> strict Subspace of V equals :Def4: the RLSStruct of V; coherence proof set W = the RLSStruct of V; W is Abelian add-associative right_zeroed right_complementable RealLinearSpace-like proof A1: 0.W = the Zero of W by RLVECT_1:def 2 .= 0.V by RLVECT_1:def 2; A2: now let a; let v,w be VECTOR of W, v',w' be VECTOR of V such that A3: v=v' & w=w'; thus v+w = (the add of W).[v,w] by RLVECT_1:def 3 .= v'+w' by A3,RLVECT_1:def 3; thus a*v = (the Mult of W).[a,v] by RLVECT_1:def 4 .= a*v' by A3,RLVECT_1:def 4; end; thus for v,w being VECTOR of W holds v + w = w + v proof let v,w be VECTOR of W; reconsider v'=v,w'=w as VECTOR of V; thus v + w = w' + v' by A2 .= w + v by A2; end; thus for u,v,w being VECTOR of W holds (u + v) + w = u + (v + w) proof let u,v,w be VECTOR of W; reconsider u'=u,v'=v,w'=w as VECTOR of V; A4: v + w = v' + w' & u + v = u' + v' by A2; hence (u + v) + w = (u' + v') + w' by A2 .= u' + (v' + w') by RLVECT_1:def 6 .= u + (v + w) by A2,A4; end; thus for v being VECTOR of W holds v + 0.W = v proof let v be VECTOR of W; reconsider v'=v as VECTOR of V; thus v + 0.W = v' + 0.V by A1,A2 .= v by RLVECT_1:10; end; thus for v being VECTOR of W ex w being VECTOR of W st v + w = 0.W proof let v be VECTOR of W; reconsider v'=v as VECTOR of V; consider w' being VECTOR of V such that A5: v' + w' = 0.V by RLVECT_1:def 8; reconsider w=w' as VECTOR of W; take w; thus v + w = 0.W by A1,A2,A5; end; thus for a for v,w being VECTOR of W holds a * (v + w) = a * v + a * w proof let a; let v,w be VECTOR of W; reconsider v'=v,w'=w as VECTOR of V; A6: v + w = v' + w' & a * v = a * v' & a * w = a * w' by A2; hence a * (v + w) = a * (v' + w') by A2 .= a * v' + a * w' by RLVECT_1:def 9 .= a * v + a * w by A2,A6; end; thus for a,b for v being VECTOR of W holds (a + b) * v = a * v + b * v proof let a,b; let v be VECTOR of W; reconsider v'=v as VECTOR of V; A7: a * v = a * v' & b * v = b * v' by A2; thus (a + b) * v = (a + b) * v' by A2 .= a * v' + b * v' by RLVECT_1:def 9 .= a * v + b * v by A2,A7; end; thus for a,b for v being VECTOR of W holds (a * b) * v = a * (b * v) proof let a,b; let v be VECTOR of W; reconsider v'=v as VECTOR of V; A8: b * v = b * v' by A2; thus (a * b) * v = (a * b) * v' by A2 .= a * (b * v') by RLVECT_1:def 9 .= a * (b * v) by A2,A8; end; thus for v being VECTOR of W holds 1 * v = v proof let v be VECTOR of W; reconsider v'=v as VECTOR of V; thus 1 * v = 1 * v' by A2 .= v by RLVECT_1:def 9; end; end; then reconsider W as RealLinearSpace; W is Subspace of V proof thus the carrier of W c= the carrier of V & the Zero of W = the Zero of V; thus thesis by FUNCT_2:40; end; hence thesis; end; end; :: :: Definitional theorems of zero subspace and improper subspace. :: canceled 4; theorem Th48: (0).W = (0).V proof the carrier of (0).W = {0.W} & the carrier of (0).V = {0.V} by Def3; then the carrier of (0).W = the carrier of (0).V & (0).W is Subspace of V by Th19, Th35; hence thesis by Th38; end; theorem Th49: (0).W1 = (0).W2 proof (0).W1 = (0).V & (0).W2 = (0).V by Th48; hence thesis; end; theorem (0).W is Subspace of V by Th35; theorem (0).V is Subspace of W proof the carrier of (0).V = {0.V} by Def3 .= {0.W} by Th19 .= {the Zero of W} by RLVECT_1:def 2; hence thesis by Th36; end; theorem (0).W1 is Subspace of W2 proof (0).W1 = (0).W2 & (0).W2 is Subspace of W2 by Th49; hence thesis; end; canceled; theorem for V being strict RealLinearSpace holds V is Subspace of (Omega).V proof let V be strict RealLinearSpace; V is Subspace of V by Th33; hence thesis by Def4; end; :: :: Introduction of the cosets of subspace. :: definition let V; let v,W; func v + W -> Subset of V equals :Def5: {v + u : u in W}; coherence proof defpred P[set] means ex u st $1 = v + u & u in W; consider X being set such that A1: for x being set holds x in X iff x in the carrier of V & P[x] from Separation; X c= the carrier of V proof let x be set; assume x in X; hence x in the carrier of V by A1; end; then reconsider X as Subset of V; set Y = {v + u : u in W}; X = Y proof thus X c= Y proof let x be set; assume x in X; then ex u st x = v + u & u in W by A1; hence thesis; end; thus Y c= X proof let x be set; assume x in Y; then ex u st x = v + u & u in W; hence thesis by A1; end; end; hence thesis; end; end; Lm2: 0.V + W = the carrier of W proof set A = {0.V + u : u in W}; A1: 0.V + W = A by Def5; A2: A c= the carrier of W proof let x be set; assume x in A; then consider u such that A3: x = 0.V + u and A4: u in W; x = u by A3,RLVECT_1:10; hence thesis by A4,RLVECT_1:def 1; end; the carrier of W c= A proof let x be set; assume x in the carrier of W; then A5: x in W by RLVECT_1:def 1; then x in V by Th17; then reconsider y = x as Element of V by RLVECT_1:def 1; 0.V + y = x by RLVECT_1:10; hence thesis by A5; end; hence thesis by A1,A2,XBOOLE_0:def 10; end; definition let V; let W; mode Coset of W -> Subset of V means :Def6: ex v st it = v + W; existence proof reconsider VW = the carrier of W as Subset of V by Def2; take VW; take 0.V; thus thesis by Lm2; end; end; reserve B,C for Coset of W; :: :: Definitional theorems of the cosets. :: canceled 3; theorem Th58: 0.V in v + W iff v in W proof set A = {v + u : u in W}; thus 0.V in v + W implies v in W proof assume 0.V in v + W; then 0.V in A by Def5; then consider u such that A1: 0.V = v + u and A2: u in W; v = - u by A1,RLVECT_1:def 10; hence thesis by A2,Th30; end; assume v in W; then A3: - v in W by Th30; 0.V = v - v by RLVECT_1:28 .= v + (- v) by RLVECT_1:def 11; then 0.V in A by A3; hence thesis by Def5; end; theorem Th59: v in v + W proof v + 0.V = v & 0.V in W by Th25,RLVECT_1:10; then v in {v + u : u in W}; hence thesis by Def5; end; theorem 0.V + W = the carrier of W by Lm2; theorem Th61: v + (0).V = {v} proof set A = {v + u : u in (0).V}; thus v + (0).V c= {v} proof let x be set; assume x in v + (0).V; then x in A by Def5; then consider u such that A1: x = v + u and A2: u in (0).V; the carrier of (0).V = {0.V} & u in the carrier of (0).V by A2,Def3,RLVECT_1:def 1 ; then u = 0.V by TARSKI:def 1; then x = v by A1,RLVECT_1:10; hence thesis by TARSKI:def 1; end; let x be set; assume x in {v}; then A3: x = v by TARSKI:def 1; 0.V in (0).V & v = v + 0.V by Th25,RLVECT_1:10; then x in A by A3; hence thesis by Def5; end; Lm3: v in W iff v + W = the carrier of W proof set A = {v + u : u in W}; thus v in W implies v + W = the carrier of W proof assume A1: v in W; thus v + W c= the carrier of W proof let x be set; assume x in v + W; then x in A by Def5; then consider u such that A2: x = v + u and A3: u in W; v + u in W by A1,A3,Th28; hence thesis by A2,RLVECT_1:def 1; end; let x be set; assume x in the carrier of W; then reconsider y = x, z = v as Element of W by A1,RLVECT_1:def 1; reconsider y1 = y, z1 = z as VECTOR of V by Th18; A4: y - z in W by RLVECT_1:def 1; A5: z + (y - z) = (y + z) - z by RLVECT_1:42 .= y + (z - z) by RLVECT_1:42 .= y + 0.W by RLVECT_1:28 .= x by RLVECT_1:10; A6: y - z = y1 - z1 by Th24; A7: y1 - z1 in W by A4,Th24; z1 + (y1 - z1) = x by A5,A6,Th21; then x in A by A7; hence thesis by Def5; end; assume A8: v + W = the carrier of W; assume A9: not v in W; 0.V in W & v + 0.V = v by Th25,RLVECT_1:10; then v in {v + u : u in W}; then v in the carrier of W by A8,Def5; hence thesis by A9,RLVECT_1:def 1; end; theorem Th62: v + (Omega).V = the carrier of V proof A1: the carrier of (Omega).V = the carrier of the RLSStruct of V by Def4 .= the carrier of V; then v in (Omega).V by RLVECT_1:def 1; hence thesis by A1,Lm3; end; theorem Th63: 0.V in v + W iff v + W = the carrier of W proof (0.V in v + W iff v in W) & (v in W iff v + W = the carrier of W) by Lm3,Th58; hence thesis; end; theorem v in W iff v + W = the carrier of W by Lm3; theorem Th65: v in W implies (a * v) + W = the carrier of W proof set A = {a * v + u : u in W}; assume A1: v in W; thus (a * v) + W c= the carrier of W proof let x be set; assume x in (a * v) + W; then x in A by Def5; then consider u such that A2: x = a * v + u and A3: u in W; a * v in W by A1,Th29; then a * v + u in W by A3,Th28; hence thesis by A2,RLVECT_1:def 1; end; let x be set; assume A4: x in the carrier of W; the carrier of W c= the carrier of V & v in V by Def2,RLVECT_1:3; then reconsider y = x as Element of V by A4; a * v in W & x in W by A1,A4,Th29,RLVECT_1:def 1; then A5: y - a * v in W by Th31; a * v + (y - a * v) = (y + a * v) - a * v by RLVECT_1:42 .= y + (a * v - a * v) by RLVECT_1:42 .= y + 0.V by RLVECT_1:28 .= x by RLVECT_1:10; then x in A by A5; hence thesis by Def5; end; theorem Th66: a <> 0 & (a * v) + W = the carrier of W implies v in W proof assume that A1: a <> 0 and A2: (a * v) + W = the carrier of W; assume not v in W; then not 1 * v in W by RLVECT_1:def 9; then not (a" * a) * v in W by A1,XCMPLX_0:def 7; then not a" * (a * v) in W by RLVECT_1:def 9; then A3: not a * v in W by Th29; 0.V in W & a * v + 0.V = a * v by Th25,RLVECT_1:10; then a * v in {a * v + u : u in W}; then a * v in the carrier of W by A2,Def5; hence contradiction by A3,RLVECT_1:def 1; end; theorem Th67: v in W iff - v + W = the carrier of W proof (v in W iff ((- 1) * v) + W = the carrier of W) & (- 1) * v = - v by Th65,Th66,RLVECT_1:29; hence thesis; end; theorem Th68: u in W iff v + W = (v + u) + W proof set A = {v + v1 : v1 in W}; set B = {(v + u) + v2 : v2 in W}; thus u in W implies v + W = (v + u) + W proof assume A1: u in W; thus v + W c= (v + u) + W proof let x be set; assume x in v + W; then x in A by Def5; then consider v1 such that A2: x = v + v1 and A3: v1 in W; A4: v1 - u in W by A1,A3,Th31; (v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def 6 .= v + ((v1 + u) - u) by RLVECT_1:42 .= v + (v1 + (u - u)) by RLVECT_1:42 .= v + (v1 + 0.V) by RLVECT_1:28 .= x by A2,RLVECT_1:10; then x in B by A4; hence thesis by Def5; end; let x be set; assume x in (v + u) + W; then x in B by Def5; then consider v2 such that A5: x = (v + u) + v2 and A6: v2 in W; A7: u + v2 in W by A1,A6,Th28; x = v + (u + v2) by A5,RLVECT_1:def 6; then x in A by A7; hence thesis by Def5; end; assume A8: v + W = (v + u) + W; 0.V in W & v + 0.V = v by Th25,RLVECT_1:10; then v in A; then v in (v + u) + W by A8,Def5; then v in B by Def5; then consider u1 such that A9: v = (v + u) + u1 and A10: u1 in W; v = v + 0.V & v = v + (u + u1) by A9,RLVECT_1:10,def 6; then A11: u + u1 = 0.V by RLVECT_1:21; u = - u1 by A11,RLVECT_1:def 10; hence thesis by A10,Th30; end; theorem u in W iff v + W = (v - u) + W proof A1: (- u in W iff v + W = (v + (- u)) + W) & v + (- u) = v - u by Th68,RLVECT_1:def 11; - u in W implies u in W proof assume - u in W; then - (- u) in W by Th30; hence thesis by RLVECT_1:30; end; hence thesis by A1,Th30; end; theorem Th70: v in u + W iff u + W = v + W proof set A = {u + v1 : v1 in W}; set B = {v + v2 : v2 in W}; thus v in u + W implies u + W = v + W proof assume v in u + W; then v in A by Def5; then consider z being VECTOR of V such that A1: v = u + z and A2: z in W; thus u + W c= v + W proof let x be set; assume x in u + W; then x in A by Def5; then consider v1 such that A3: x = u + v1 and A4: v1 in W; A5: v1 - z in W by A2,A4,Th31; v - z = u + (z - z) by A1,RLVECT_1:42 .= u + 0.V by RLVECT_1:28 .= u by RLVECT_1:10; then x = (v + (- z)) + v1 by A3,RLVECT_1:def 11 .= v + (v1 + (- z)) by RLVECT_1:def 6 .= v + (v1 - z) by RLVECT_1:def 11; then x in B by A5; hence thesis by Def5; end; let x be set; assume x in v + W; then x in B by Def5; then consider v2 such that A6: x = v + v2 and A7: v2 in W; A8: z + v2 in W by A2,A7,Th28; x = u + (z + v2) by A1,A6,RLVECT_1:def 6; then x in A by A8; hence thesis by Def5; end; thus thesis by Th59; end; theorem Th71: v + W = (- v) + W iff v in W proof thus v + W = (- v) + W implies v in W proof assume v + W = (- v) + W; then v in (- v) + W by Th59; then v in {- v + u : u in W} by Def5; then consider u such that A1: v = - v + u and A2: u in W; 0.V = v - (- v + u) by A1,RLVECT_1:28 .= (v - (- v)) - u by RLVECT_1:41 .= (v + (- (- v))) - u by RLVECT_1:def 11 .= (v + v) - u by RLVECT_1:30 .= (1 * v + v) - u by RLVECT_1:def 9 .= (1 * v + 1 * v) - u by RLVECT_1:def 9 .= ((1 + 1) * v) - u by RLVECT_1:def 9 .= 2 * v - u; then 2" * (2 * v) = 2" * u by RLVECT_1:35; then (2" * 2) * v = 2" * u & 0 <> 2 by RLVECT_1:def 9; then v = 2" * u by RLVECT_1:def 9; hence thesis by A2,Th29; end; assume v in W; then v + W = the carrier of W & (- v) + W = the carrier of W by Lm3,Th67; hence thesis; end; theorem Th72: u in v1 + W & u in v2 + W implies v1 + W = v2 + W proof assume that A1: u in v1 + W and A2: u in v2 + W; set A = {v1 + u1 : u1 in W}; set B = {v2 + u2 : u2 in W}; u in A by A1,Def5; then consider x1 being VECTOR of V such that A3: u = v1 + x1 and A4: x1 in W; u in B by A2,Def5; then consider x2 being VECTOR of V such that A5: u = v2 + x2 and A6: x2 in W; thus v1 + W c= v2 + W proof let x be set; assume x in v1 + W; then x in A by Def5; then consider u1 such that A7: x = v1 + u1 and A8: u1 in W; u - x1 = v1 + (x1 - x1) by A3,RLVECT_1:42 .= v1 + 0.V by RLVECT_1:28 .= v1 by RLVECT_1:10; then A9: x = (v2 + (x2 - x1)) + u1 by A5,A7,RLVECT_1:42 .= v2 + ((x2 - x1) + u1) by RLVECT_1:def 6; x2 - x1 in W by A4,A6,Th31; then (x2 - x1) + u1 in W by A8,Th28; then x in B by A9; hence thesis by Def5; end; let x be set; assume x in v2 + W; then x in B by Def5; then consider u1 such that A10: x = v2 + u1 and A11: u1 in W; u - x2 = v2 + (x2 - x2) by A5,RLVECT_1:42 .= v2 + 0.V by RLVECT_1:28 .= v2 by RLVECT_1:10; then A12: x = (v1 + (x1 - x2)) + u1 by A3,A10,RLVECT_1:42 .= v1 + ((x1 - x2) + u1) by RLVECT_1:def 6; x1 - x2 in W by A4,A6,Th31; then (x1 - x2) + u1 in W by A11,Th28; then x in A by A12; hence thesis by Def5; end; theorem u in v + W & u in (- v) + W implies v in W proof assume u in v + W & u in (- v) + W; then v + W = (- v) + W by Th72; hence thesis by Th71; end; theorem Th74: a <> 1 & a * v in v + W implies v in W proof assume that A1: a <> 1 and A2: a * v in v + W; A3: now assume a - 1 = 0; then (- 1) + a = 0 by XCMPLX_0:def 8; then a = - (- 1) by XCMPLX_0:def 6; hence contradiction by A1; end; a * v in {v + u : u in W} by A2,Def5; then consider u such that A4: a * v = v + u and A5: u in W; u = u + 0.V by RLVECT_1:10 .= u + (v - v) by RLVECT_1:28 .= a * v - v by A4,RLVECT_1:42 .= a * v - 1 * v by RLVECT_1:def 9 .= (a - 1) * v by RLVECT_1:49; then (a - 1)" * u = ((a - 1)" * (a - 1)) * v & a - 1 <> 0 by A3,RLVECT_1:def 9; then 1 * v = (a - 1)" * u by XCMPLX_0:def 7; then v = (a - 1)" * u by RLVECT_1:def 9; hence thesis by A5,Th29; end; theorem Th75: v in W implies a * v in v + W proof assume A1: v in W; A2: a * v = (a - (1 - 1)) * v .= ((a - 1) + 1) * v by XCMPLX_1:37 .= (a - 1) * v + 1 * v by RLVECT_1:def 9 .= v + (a - 1) * v by RLVECT_1:def 9; (a - 1) * v in W by A1,Th29; then a * v in {v + u : u in W} by A2; hence thesis by Def5; end; theorem - v in v + W iff v in W proof (v in W implies (- 1) * v in v + W) & (- 1) * v = - v & (-1 <> 1 & (- 1) * v in v + W implies v in W) by Th74,Th75,RLVECT_1:29; hence thesis; end; theorem Th77: u + v in v + W iff u in W proof set A = {v + v1 : v1 in W}; thus u + v in v + W implies u in W proof assume u + v in v + W; then u + v in A by Def5; then consider v1 such that A1: u + v = v + v1 and A2: v1 in W; thus thesis by A1,A2,RLVECT_1:21; end; assume u in W; then u + v in A; hence thesis by Def5; end; theorem v - u in v + W iff u in W proof A1: v - u = (- u) + v by RLVECT_1:def 11; A2: u in W implies - u in W by Th30; - u in W implies - (- u) in W by Th30; hence thesis by A1,A2,Th77,RLVECT_1:30; end; theorem Th79: u in v + W iff (ex v1 st v1 in W & u = v + v1) proof set A = {v + v2 : v2 in W}; thus u in v + W implies (ex v1 st v1 in W & u = v + v1) proof assume u in v + W; then u in A by Def5; then ex v1 st u = v + v1 & v1 in W; hence thesis; end; given v1 such that A1: v1 in W & u = v + v1; u in A by A1; hence thesis by Def5; end; theorem u in v + W iff (ex v1 st v1 in W & u = v - v1) proof set A = {v + v2 : v2 in W}; thus u in v + W implies (ex v1 st v1 in W & u = v - v1) proof assume u in v + W; then u in A by Def5; then consider v1 such that A1: u = v + v1 and A2: v1 in W; take x = - v1; thus x in W by A2,Th30; u = v + (- (- v1)) by A1,RLVECT_1:30 .= v - (- v1) by RLVECT_1:def 11; hence thesis; end; given v1 such that A3: v1 in W & u = v - v1; u = v + (- v1) & - v1 in W by A3,Th30,RLVECT_1:def 11; then u in A; hence thesis by Def5; end; theorem Th81: (ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W proof thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W proof given v such that A1: v1 in v + W and A2: v2 in v + W; consider u1 such that A3: u1 in W and A4: v1 = v + u1 by A1,Th79; consider u2 such that A5: u2 in W and A6: v2 = v + u2 by A2,Th79; v1 - v2 = (u1 + v) + (- (v + u2)) by A4,A6,RLVECT_1:def 11 .= (u1 + v) + ((- v) - u2) by RLVECT_1:44 .= ((u1 + v) + (- v)) - u2 by RLVECT_1:42 .= (u1 + (v + (- v))) - u2 by RLVECT_1:def 6 .= (u1 + 0.V) - u2 by RLVECT_1:16 .= u1 - u2 by RLVECT_1:10; hence thesis by A3,A5,Th31; end; assume v1 - v2 in W; then A7: - (v1 - v2) in W by Th30; take v1; thus v1 in v1 + W by Th59; v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by RLVECT_1:47 .= (v1 + (- v1)) + v2 by RLVECT_1:def 6 .= 0.V + v2 by RLVECT_1:16 .= v2 by RLVECT_1:10; hence thesis by A7,Th79; end; theorem Th82: v + W = u + W implies (ex v1 st v1 in W & v + v1 = u) proof assume A1: v + W = u + W; take v1 = u - v; v in u + W by A1,Th59; then v in {u + u2 : u2 in W} by Def5; then consider u1 such that A2: v = u + u1 and A3: u1 in W; 0.V = (u + u1) - v by A2,RLVECT_1:28 .= u + (u1 - v) by RLVECT_1:42 .= u + ((- v) + u1) by RLVECT_1:def 11 .= (u + (- v)) + u1 by RLVECT_1:def 6 .= u1 + (u - v) by RLVECT_1:def 11; then v1 = - u1 by RLVECT_1:def 10; hence v1 in W by A3,Th30; thus v + v1 = (u + v) - v by RLVECT_1:42 .= u + (v - v) by RLVECT_1:42 .= u + 0.V by RLVECT_1:28 .= u by RLVECT_1:10; end; theorem Th83: v + W = u + W implies (ex v1 st v1 in W & v - v1 = u) proof assume A1: v + W = u + W; take v1 = v - u; u in v + W by A1,Th59; then u in {v + u2 : u2 in W} by Def5; then consider u1 such that A2: u = v + u1 and A3: u1 in W; 0.V = (v + u1) - u by A2,RLVECT_1:28 .= v + (u1 - u) by RLVECT_1:42 .= v + ((- u) + u1) by RLVECT_1:def 11 .= (v + (- u)) + u1 by RLVECT_1:def 6 .= u1 + (v - u) by RLVECT_1:def 11; then v1 = - u1 by RLVECT_1:def 10; hence v1 in W by A3,Th30; thus v - v1 = (v - v) + u by RLVECT_1:43 .= 0.V + u by RLVECT_1:28 .= u by RLVECT_1:10; end; theorem Th84: for W1,W2 being strict Subspace of V holds v + W1 = v + W2 iff W1 = W2 proof let W1,W2 be strict Subspace of V; thus v + W1 = v + W2 implies W1 = W2 proof assume A1: v + W1 = v + W2; the carrier of W1 = the carrier of W2 proof A2: the carrier of W1 c= the carrier of V by Def2; A3: the carrier of W2 c= the carrier of V by Def2; thus the carrier of W1 c= the carrier of W2 proof let x be set; assume A4: x in the carrier of W1; then reconsider y = x as Element of V by A2; set z = v + y; x in W1 by A4,RLVECT_1:def 1; then z in {v + u : u in W1}; then z in v + W2 by A1,Def5; then z in {v + u : u in W2} by Def5; then consider u such that A5: z = v + u and A6: u in W2; y = u by A5,RLVECT_1:21; hence thesis by A6,RLVECT_1:def 1; end; let x be set; assume A7: x in the carrier of W2; then reconsider y = x as Element of V by A3; set z = v + y; x in W2 by A7,RLVECT_1:def 1; then z in {v + u : u in W2}; then z in v + W1 by A1,Def5; then z in {v + u : u in W1} by Def5; then consider u such that A8: z = v + u and A9: u in W1; y = u by A8,RLVECT_1:21; hence thesis by A9,RLVECT_1:def 1; end; hence thesis by Th38; end; thus thesis; end; theorem Th85: for W1,W2 being strict Subspace of V holds v + W1 = u + W2 implies W1 = W2 proof let W1,W2 be strict Subspace of V; assume A1: v + W1 = u + W2; set V1 = the carrier of W1; set V2 = the carrier of W2; assume A2: W1 <> W2; then V1 <> V2 by Th38; then A3: not V1 c= V2 or not V2 c= V1 by XBOOLE_0:def 10; A4: now assume A5: V1 \ V2 <> {}; consider x being Element of V1 \ V2; x in V1 & not x in V2 by A5,XBOOLE_0:def 4; then A6: x in W1 & not x in W2 by RLVECT_1:def 1; then x in V by Th17; then reconsider x as Element of V by RLVECT_1:def 1; set z = v + x; z in {v + u2 : u2 in W1} by A6; then z in u + W2 by A1,Def5; then z in {u + u2 : u2 in W2} by Def5; then consider u1 such that A7: z = u + u1 and A8: u1 in W2; x = 0.V + x by RLVECT_1:10 .= v - v + x by RLVECT_1:28 .= (- v + v) + x by RLVECT_1:def 11 .= - v + (u + u1) by A7,RLVECT_1:def 6; then A9: (v + (- v + (u + u1))) + W1 = v + W1 by A6,Th68; v + (- v + (u + u1)) = (v + (- v)) + (u + u1) by RLVECT_1:def 6 .= (v - v) + (u + u1) by RLVECT_1:def 11 .= 0.V + (u + u1) by RLVECT_1:28 .= u + u1 by RLVECT_1:10; then (u + u1) + W2 = (u + u1) + W1 by A1,A8,A9,Th68; hence thesis by A2,Th84; end; now assume A10: V2 \ V1 <> {}; consider x being Element of V2 \ V1; x in V2 & not x in V1 by A10,XBOOLE_0:def 4; then A11: x in W2 & not x in W1 by RLVECT_1:def 1; then x in V by Th17; then reconsider x as Element of V by RLVECT_1:def 1; set z = u + x; z in {u + u2 : u2 in W2} by A11; then z in v + W1 by A1,Def5; then z in {v + u2 : u2 in W1} by Def5; then consider u1 such that A12: z = v + u1 and A13: u1 in W1; x = 0.V + x by RLVECT_1:10 .= u - u + x by RLVECT_1:28 .= (- u + u) + x by RLVECT_1:def 11 .= - u + (v + u1) by A12,RLVECT_1:def 6; then A14: (u + (- u + (v + u1))) + W2 = u + W2 by A11,Th68; u + (- u + (v + u1)) = (u + (- u)) + (v + u1) by RLVECT_1:def 6 .= (u - u) + (v + u1) by RLVECT_1:def 11 .= 0.V + (v + u1) by RLVECT_1:28 .= v + u1 by RLVECT_1:10; then (v + u1) + W1 = (v + u1) + W2 by A1,A13,A14,Th68; hence thesis by A2,Th84; end; hence thesis by A3,A4,XBOOLE_1:37; end; :: :: Theorems concerning cosets of subspace :: regarded as subsets of the carrier. :: theorem C is lineary-closed iff C = the carrier of W proof thus C is lineary-closed implies C = the carrier of W proof assume A1: C is lineary-closed; consider v such that A2: C = v + W by Def6; C <> {} by A2,Th59; then 0.V in v + W by A1,A2,Th4; hence thesis by A2,Th63; end; thus thesis by Lm1; end; theorem for W1,W2 being strict Subspace of V, C1 being Coset of W1, C2 being Coset of W2 holds C1 = C2 implies W1 = W2 proof let W1,W2 be strict Subspace of V, C1 be Coset of W1, C2 be Coset of W2; A1: ex v1 st C1 = v1 + W1 by Def6; ex v2 st C2 = v2 + W2 by Def6; hence thesis by A1,Th85; end; theorem {v} is Coset of (0).V proof v + (0).V = {v} by Th61; hence thesis by Def6; end; theorem V1 is Coset of (0).V implies (ex v st V1 = {v}) proof assume V1 is Coset of (0).V; then consider v such that A1: V1 = v + (0).V by Def6; take v; thus thesis by A1,Th61; end; theorem the carrier of W is Coset of W proof the carrier of W = 0.V + W by Lm2; hence thesis by Def6; end; theorem the carrier of V is Coset of (Omega).V proof the carrier of V is Subset of V iff the carrier of V c= the carrier of V; then reconsider A = the carrier of V as Subset of V; consider v; A = v + (Omega).V by Th62; hence thesis by Def6; end; theorem V1 is Coset of (Omega).V implies V1 = the carrier of V proof assume V1 is Coset of (Omega).V; then ex v st V1 = v + (Omega).V by Def6; hence thesis by Th62; end; theorem 0.V in C iff C = the carrier of W proof ex v st C = v + W by Def6; hence thesis by Th63; end; theorem Th94: u in C iff C = u + W proof thus u in C implies C = u + W proof assume A1: u in C; ex v st C = v + W by Def6; hence thesis by A1,Th70; end; thus thesis by Th59; end; theorem u in C & v in C implies (ex v1 st v1 in W & u + v1 = v) proof assume u in C & v in C; then C = u + W & C = v + W by Th94; hence thesis by Th82; end; theorem u in C & v in C implies (ex v1 st v1 in W & u - v1 = v) proof assume u in C & v in C; then C = u + W & C = v + W by Th94; hence thesis by Th83; end; theorem (ex C st v1 in C & v2 in C) iff v1 - v2 in W proof thus (ex C st v1 in C & v2 in C) implies v1 - v2 in W proof given C such that A1: v1 in C & v2 in C; ex v st C = v + W by Def6; hence thesis by A1,Th81; end; assume v1 - v2 in W; then consider v such that A2: v1 in v + W & v2 in v + W by Th81; reconsider C = v + W as Coset of W by Def6; take C; thus thesis by A2; end; theorem u in B & u in C implies B = C proof assume A1: u in B & u in C; A2: ex v1 st B = v1 + W by Def6; ex v2 st C = v2 + W by Def6; hence thesis by A1,A2,Th72; end;

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